Perpendicular lines

Activity 1: Investigation: Perpendicular lines

Draw a sketch of the line passing through the points \(A(-2;-3)\) and \(B(2;5)\) and the line passing through the points \(C(-1;\frac{1}{2})\) and \(D(4;-2)\).

Label and measure \(\alpha\) and \(\beta\), the angles of inclination of straight lines \(AB\) and \(CD\) respectively.

Label and measure \(\theta\), the angle between the lines \(AB\) and \(CD\).

Describe the relationship between the lines \(AB\) and \(CD\).

“\(\theta\) is a reflex angle, therefore \(AB \perp CD\).” Is this a true statement? If not, provide a correct statement.

Determine the equation of the straight line \(AB\) and the line \(CD\).

Use your calculator to determine \(\tan \alpha \times \tan \beta\).

Determine \(m_{AB} \times m_{CD}\).

What do you notice about these products?

Complete the sentence: if two lines are \(\ldots \ldots\) to each other, then the product of their \(\ldots \ldots\) is equal \(\ldots \ldots\)

Complete the sentence: if the gradient of a straight line is equal to the negative \(\ldots \ldots\) of the gradient of another straight line, then the two lines are \(\ldots \ldots\)

Deriving the formula: \(m_1 \times m_2 = -1\)

Consider the point \(A(4;3)\) on the Cartesian plane with an angle of inclination \(A\hat{O}X = \theta\). Rotate through an angle of \(\textrm{90}\text{°}\) and place point \(B\) at \((-3;4)\) so that we have the angle of inclination \(B\hat{O}X = \textrm{90}\text{°}+ \theta\).

Another method of determining the equation of a straight line is to be given a point on the line, \(\left({x}_{1};{y}_{1}\right)\), and the equation of a line which is perpendicular to the unknown line. Let the equation of the unknown line be \(y = m_1x + c_1\) and the equation of the given line be \(y = m_2x + c_2\).

If the two lines are perpendicular then

\[m_1 \times m_2 = -1\]

Note: this rule does not apply to vertical or horizontal lines.

When determining the gradient of a line using the coefficient of \(x\), make sure the given equation is written in the gradient–intercept (standard) form \(y = mx + c\). Then we know that \[m_1 = -\frac{1}{m_2}\]

Substitute the value of \(m_1\) and the given point \(\left({x}_{1};{y}_{1}\right)\), into the gradient–intercept form of the straight line equation \(y-y_1 = m(x-x_1)\) and determine the equation of the unknown line.

Example 1: Perpendicular lines

Question

Determine the equation of the straight line passing through the point \(T(2;2)\) and perpendicular to the line \(3y + 2x - 6 = 0\).

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